Download January 2010

Part I - Mechanics
J10M.1 - Rod on a Rail (M93M.2)
J10M.1 - Rod on a Rail (M93M.2)
Problem
α
s
`
θ
g
z
x
A uniform rod of length ` and mass m moves in the x-z plane. One end of the rod is suspended
from a straight rail that slopes downwards with an angle α relative to the horizontal; the connection
point is free to move along the rail without friction, and the rod is able to swing freely in the x-z
plane. Uniform gravity acts downwards.
a)
Construct the Lagrangian of this system in terms of generalized coordinates s (the distance the
connection point has moved along the rail) and θ (the angle the rod makes with the vertical
direction).
b)
Using your Lagrangian, find a solution to the equation of motion where the rod moves with
fixed θ as s increases.
c)
Explain how your solution is consistent with (and can be derived from) the equivalence principle.
Part I - Mechanics
J10M.2 - Orbiting Mass on a String (J00M.3)
J10M.2 - Orbiting Mass on a String (J00M.3)
Problem
A hockey puck with mass m can move without friction or air resistance on the smooth horizontal
surface of a table. A massless string attached to the puck passes through a hole in the table (through
which it can slide without friction) and a mass M is suspended from its other end. Gravity acts
uniformly in the downward direction. Treat the puck as a point mass.
a)
~ 0 of the puck relative to the hole,
Given the masses m and M , plus the initial displacement R
and its initial velocity ~v0 in the plane of the table surface, find the equation that determines
the maximum and minimum radial distances of the puck from the hole during its orbit. (Don’t
bother to solve this equation!)
b)
Find the frequency of oscillations of the radial distance when the orbit is close to being circular.
Part I - Mechanics
J10M.3 - Slab on Rotating Rollers
J10M.3 - Slab on Rotating Rollers
Problem
g
x
M
d
A uniform rigid slab of mass M is supported by two rapidly counter-rotating parallel horizontal
rollers, with axes a distance d apart, with surfaces that brush past the slab in the directions shown
in the figure. The coefficient of kinetic friction between each roller and the slab is µk .
At time t = 0, the center of mass of the slab is initially displaced horizontally by x(0) = x0 (where
|x0 | < d/2) relative to the midpoint between the rollers, and the slab is initially at rest, ẋ(0) = 0.
a)
Write down the equation of motion for x(t), and solve it for t > 0 with the given initial
conditions.
Now consider the case where the directions of the rollers are reversed, as shown below:
b)
Calculate x(t) for t > 0 for the same initial conditions, in this second case.
Part II - E & M
J10E.1 - Oscillating Dipole Near a Conducting Plane
J10E.1 - Oscillating Dipole Near a Conducting Plane
Problem
oscillating dipole
p~0 cos ω0 t
d λ0
conducting plane
An electric dipole is forced to oscillate with frequency ω0 and amplitude p~0 , so p~(t) = p~0 cos ωt. It
is placed in vacuum at a distnace d c/ω0 = λ0 away from an infinite perfectly-conducting plane,
with p~0 parallel to the plane. The physical dimensions of the dipole are infinitesimal compared to
compared to d, and it can be treated as a point dipole.
At distances from the dipole that are large compared to λ0 :
a)
~ r, t) and B(~
~ r, t).
Find the steady-state electromagnetic fields E(~
b)
Find the angular distribution of the radiated power of the emitted radiation.
Part II - E & M
J10E.2 - Rotating Sphere in a Magnetic Field
J10E.2 - Rotating Sphere in a Magnetic Field
Problem
ω
~0
B
a
A solid metallic sphere of radius a has finite conductivity, carries no net electric charge, and is free
to rotate without friction about a vertical axis through its center. The region outside the sphere is
~ 0 parallel to the axis.
vacuum. There is a uniform magnetic field with flux density B
The sphere is given an impulse that starts it spinning around the axis and there is some initial
Ohmic dissipation. After the dissipation has ceased, the sphere is in a steady state of rigid rotation
with constant angular velocity ω∞ .
In steady state, to lowest order in both B0 and ω∞ , find:
a)
~ r) and electric potential Φ(~r) in the interior of the sphere, r < a. (Give
The electric field E(~
these in the non-rotating “laboratory frame”.)
b)
The electric potential outside the sphere. (Express your answer in spherical coordinates
(r, θ, φ).) State the nature of the electric field it describes (i.e., monoole, dipole, quadrupole,
etc.).
c)
The induced bulk and surface charge density distributions in the conductor that give rise to
this potential.
Note: By working to lowest order in B0 and ω∞ , you can ignore both the mechanical deformation of
the metal sphere due to rotation and the magnetic fields generated by currents in the metal (therse
are negligibly small relative to B0 ).
Part II - E & M
J10E.3 - Rectangular Waveguide
J10E.3 - Rectangular Waveguide
Problem
x
z
y
a
b
A transverse electric (T.E.) wave is propagating in an infinitely long rectangular waveguide with
perfectly conducting walls. The waveguide is filled with a dielectric (dielectric constant and
relative magnetic permeability mu = 1). The electric field inside it is
πy Ex = E0 sin
ei(kz−ωt) , Ey = 0 .
b
a)
~ field.
Find the corresponding B
Suppose now that the dielectric is removed from the region z > 0 inside the waveguide, so it is
vacuum. The region z < 0 remains filled with dielectric, as before, and the electric field of the
incident wave in the region z < 0 is that given above.
b)
~ field in the vacuum region z > 0.
Find the transmitted E
c)
For what range of ω will there be no transmitted propagating wave in the vacuum region
z > 0?
Part III - Quantum
J10Q.1 - Harmonic Oscillator
J10Q.1 - Harmonic Oscillator
Problem
Consider an isotropic three-dimensional harmonic oscillator described by the rotationally-invariant
Hamiltonian
H=
|~p|2 mω 2 2
+
|~v | .
2m
2
a)
b)
i.
What are the energies and degeneracies of the lowest three energy levels?
ii.
Account for the degeneracies by classifying states in these levels into total angular momentum multiplets.
By how much does the ground state energy change under the influence of a perturbation of
the form
H 0 = λ(~b · ~x)3
where ~b is some fixed vector, and λ is small? Calculate the correction up to second order in λ.
Now suppose that the oscillating particle has charge q. At time t = 0, a weak uniform electric field
~ is switched on, which then slowly decays as E(t)
~
~ 0 e−t/τ , with τ > 0.
E
=E
c)
~ 0 |) that a system originally in the ground state
What is the probability (to leading order in |E
will be in an excited state at a much larger time t τ ?
Part III - Quantum
J10Q.2 - Angular Momentum
J10Q.2 - Angular Momentum
Problem
A two-particle system is in a state |Ψ0 i, where each particle has orbital angular momentum quantum
numbers ` = 1 and m` = 0.
~ tot = L
~1 + L
~ 2 be the total angular momentum of the two particles, where L2 has eigenvalues
Let L
tot
~2 L(L + 1).
a)
If the two-particle state is expanded in eigenstates of L2tot , which values of L have non-zero
amplitude in the expansion? For each of these values, what is the probability that it will be
~ tot |2 ?
found in a measurement of |L
At time t = 0, a coupling between the particles is “switched on”, so that for t > 0 the time evolution
of the state is governed by the Hamiltonian
~1 · L
~2 .
H = γL
The amplitude f (t) = |hΨ (t)|Ψ0 i|2 oscillates as a function of time, returning to the value 1 at times
t = tn = nT . What is the period T ?
b)
What is the value of f (t) when t = (tn + tn+1 )/2?
Part III - Quantum
J10Q.3 - Spin-Dependent Scattering
J10Q.3 - Spin-Dependent Scattering
Problem
Consider the usual Hamiltonian for non-relativistic electrons moving in 2D:
p2y
p2x
H = H0 + V (x, y) , where H0 =
+
.
2m 2m
Electrons experience a “step” potential V (x, y) = 0 for x < 0, V = V0 > 0 fro x > 0.
a)
Electrons arriving from the region x < 0 are incident normally in the step (i.e., have conserved
momentum py = 0). Find the probability of reflection.
Now consider a similar problem, but this time one where the Hamiltonian couples the spatial and
spin degrees of freedom of the electron in an essential way:
H = H0 + V (x, y)σ 0 ,
H0 = vF (σ x py − σ y px ) ,
where
where σ i are 2 × 2 Pauli matrices and σ 0 is the identity matrix; vF is a characteristic speed, and
V (x, y) is the same “step” potential as in a), above.
In this problem, eigenstates of H0 with momentum p have energie ε± (p) = ±vF |p|. They are
non-degenerate for p 6= 0.
b)
Find the two-component wavefunction Ψσ (x, y) of positive energy eigenstates of H0 with momentum p ≡ (px , py ) = |p|(cos θ, sin θ). (The index σ labels the two possible values of the
z-component of spin.)
c)
Electrons arriving from the region x < 0 with py = 0 (as in part a)) now have 100% probability
of transmission through the step. Explain why.
d)
Consider electrons arriving with energy E = 2V0 and py = |p| sin θ:
i.
For what range of θ is transmission through the step possible? (Hint: an analog of “Snell’s
law” relates angles of incidence and refraction, θ and θ0 .)
ii.
In this range, find the reflection probability R(θ).
Note 1: The second Hamiltonian only requires that both wavefunction components Ψσ (x, y), σ =↑, ↓,
are continuous at the step, with no condition on their derivatives. (It describes electrons on the
surface of a “topological insulator”.)
Note 2: The Pauli matrices are:
0 1
0 −i
x
y
σ =
, σ =
,
1 0
i 0
1 0
σ =
,
0 −1
z
1 0
σ =
.
0 1
0
Part IV - Stat Mech & Thermo
J10T.1 - Graphene
J10T.1 - Graphene
Problem
Graphene is a two-dimensional sheet of carbon atoms. Both electronic and phonon degrees of
freedom contribute to the low-temperature specific heat per unit area. The electron states resembe
the states of the massless Dirac equation, with energies
ε± (~p) = ε0 ± vF p ,
p ≡ |~p| ,
where P~ = (px , py ) is the analog of the momentum carried by a Dirac electron. (There are two
energy bands, ε+ (~p) ≥ 0 and ε− (~p) ≤ 0 which become degenerate at p~ = 0). These states have a
fourfold degeneracy (the usual two-fold spin degeneracy is doubled by an additional “valley” index).
a)
If the Fermi energy EF is ε0 + vF pF , with pF > 0, what is the leading behavior of the electronic
specific heat as T → 0?
b)
What is the low-temperature electronic specific heat when pF = 0?
(The next calculation is independent of parts a), b) above).
Recently, freely suspended graphene sheets have been studied. These have an unusual phonon
spectrum: in addition to longitudinal and transverse sound waves with frequencies ω = vL q, vT q
(where q is the magnitude of the wavenumber ~q), there is an extra low-frequency mode ω = Kq 2
where atomic displacements are normal to the sheet.
c)
Obtain the leading behavior of the phonon contribution to be specific heat as T → 0.
You may express your answers in terms of the numerical constants
Z ∞
xn
±
Cn =
dx x
, n > 0.
e ±1
0
Part IV - Stat Mech & Thermo
J10T.2 - Maxwell-Boltzmann Gas
J10T.2 - Maxwell-Boltzmann Gas
Problem
In a simple approximation often used to calculate transport properties, the statistical distribution
of the velocities of molecules arriving at a point is taken to be that of the local equilibrium state
where their most-recent collision occurred (T, p, h~v i, etc., are assumed to be slowly varying functions
of position).
a)
Using this approximation, derive the well-known estimate (due to Maxwell) of the viscosity
η of a dilute classical gas of molecules with mass m, particle density n̄, and mean free path
` between collisions. Assume a Maxwell-Boltzmann distribution of molecular velocities with
2
.
h|~v − h~v i|2 i = vrms
b)
If ` is modeled by treating monoatomic molecules as hard spheres with a finite diameter, how
does the predicted viscosity vary with pressure p for low pressures at fixed temperature T ?
(Assume that ` remains smaller than the dimensions of the container.)
The Maxwell-Boltzmann gas can be viewed as the high-temperature limit of a quantum gas of
non-relativistic particles. The Maxwell-Boltzmann treatment assumes that
λ(T ) n̄−1/3 ` ,
where λ(T ) is the thermal de Broglie wavelength of the particles.
c)
d)
In terms just of the three lengths λ(T ), n̄−1/3 , and `, plus fundamental constants, give expressions for:
i.
The viscosity η of a Maxwell-Boltzmann gas.
ii.
The entropy density s̄ of a monoatomic Maxwell-Boltzmann gas.
Estimate the lowest value that the ratio η/s̄ can take before the quantum effects neglected in
Maxwell-Boltzmann theory must be considered.
Part IV - Stat Mech & Thermo
J10T.3 - Thermodynamics of Superconductors (O85T.1)
J10T.3 - Thermodynamics of Superconductors (O85T.1)
Problem
H
Hc (T )
normal
Hc (0)
superconducting
Tc
T
In the absence of a magnetic field (H = 0), an isotropic metal has a continuous transition to
a superconducting state below a critical temperature Tc . The metal has specific heat (per unit
volume) cnV = γT , while in the superconductor csV = αT 3 . Assume that the volume of the material
does not vary with the temperature and magnetic field.
a)
Find Tc as a function of γ and α.
b)
For H = 0, give expressions in terms of T, Tc , and γ for (and sketch versus T ):
i.
the free energy density,
ii.
the entropy density
iii.
the specific heat.
In finite magnetic field strength H > 0, the transition becomes first order. The superconductor
exhibits the Meissner effect which excludes magnetic flux density B from its interior, so B = 0
even though H > 0. Above a critical field Hc (T ), superconductivity breaks down, and the system
becomes normal with B = µH (to a good approximation, µ is equal to the vacuum permeability).
The phase diagram is depicted above.
c)
On general grounds, why must dHc (T )/dT vanish as T → 0?
d)
Find an expression for Hc (T ). (Assume that cnV and csV do not depend on H.)
Note: When the internal energy U is defined to include the integrated electromagnetic energy
density inside the material, H is a thermodynamic analog of the pressure:
H ≡ V −1 ∂U/∂B|SV N .